=weight of oranges in a 5-lb bag ~ N(m,0.2); (a) How many customers get cheated, if E(X)=5? (proportion p=________?) (b) What should m equal so that (a) p<16%, (b) 2.5%, (c) 0.5% ? (d) If the cost of oranges to the packer is $0.35, what is the respective cost of meeting these constraints in a batch of 200 bags?
2. X=passengers on the 7am shuttle to DC ~ N(130,20) [X rounded to nearest integer] (a) How many customers get bumped if the capacity of the aircraft is 160? Show expression_________________ ; If the marginal profit is $100/riding passenger and the cost is $40/bumped passenger, then what is the profit if: (b) X=180 (c) X=200 (d) X=220 (e) what is the expected profit of this flight?
To answer these questions, we will need to use probability and the properties of the normal distribution. Let's go step by step.
Question 1:
(a) We are given that the weight of oranges in a 5-lb bag follows a normal distribution N(m, 0.2). We are also given that the expected value of X (the weight of oranges) is 5 lbs. In this case, since the bag weight is normally distributed, we can calculate the probability of a bag weighing less than 5 lbs or more than 5 lbs.
To find the proportion of customers who get cheated, we need to calculate the probability that a bag weighs less than 5 lbs or more than 5 lbs. This can be written as P(X < 5) + P(X > 5).
(b) In order to find the value of m that satisfies the given proportions, we need to calculate the Z-scores associated with each proportion and then use the Z-table to find the corresponding values.
The Z-score formula is: Z = (X - m) / σ, where X is the given value (in this case, the proportion), m is the mean, and σ is the standard deviation.
To solve for m:
(i) For p < 16%: Find the Z-score corresponding to p = 16% (which is 0.16), and rearrange the Z-score formula to solve for m: m = X - (Z * σ).
(ii) For p = 2.5% and p = 0.5%: Repeat the same steps as above.
(c) To calculate the cost of meeting these constraints in a batch of 200 bags, we need to multiply the number of bags that do not meet the constraints (cheated customers) in each case by the cost per bag. In this case, the cost is given as $0.35 per bag.
Question 2:
(a) We are given that the number of passengers on the shuttle to DC, X, follows a normal distribution N(130, 20), rounded to the nearest integer. We are also given that the capacity of the aircraft is 160 passengers.
To determine how many customers get bumped, we need to find the probability that X is greater than 160. This can be written as P(X > 160).
(b) For profit calculation, we are given that the marginal profit per riding passenger is $100 and the cost per bumped passenger is $40.
To calculate the profit, we need to subtract the cost for each bumped passenger from the overall profit for each case.
(c) Repeat the same steps as above for X = 200 and X = 220.
(e) To calculate the expected profit for this flight, we multiply the probability of each scenario by the profit for that scenario, and then sum them up.
Please note that the calculations for this exercise involve using the properties of the normal distribution and relevant formulas. You can utilize statistical software or tables to find precise values for the probabilities and perform the calculations accurately.